Universality for random tensors

Razvan Gurau

Annales de l'I.H.P. Probabilités et statistiques (2014)

  • Volume: 50, Issue: 4, page 1474-1525
  • ISSN: 0246-0203

Abstract

top
We prove two universality results for random tensors of arbitrary rank . We first prove that a random tensor whose entries are independent, identically distributed, complex random variables converges in distribution in the large limit to the same limit as the distributional limit of a Gaussian tensor model. This generalizes the universality of random matrices to random tensors. We then prove a second, stronger, universality result. Under the weaker assumption that the joint probability distribution of tensor entries is invariant, assuming that the cumulants of this invariant distribution are uniformly bounded, we prove that in the large limit the tensor again converges in distribution to the distributional limit of a Gaussian tensor model. We emphasize that the covariance of the large Gaussian isnot universal, but depends strongly on the details of the joint distribution.

How to cite

top

Gurau, Razvan. "Universality for random tensors." Annales de l'I.H.P. Probabilités et statistiques 50.4 (2014): 1474-1525. <http://eudml.org/doc/272101>.

@article{Gurau2014,
abstract = {We prove two universality results for random tensors of arbitrary rank $D$. We first prove that a random tensor whose entries are $N^\{D\}$ independent, identically distributed, complex random variables converges in distribution in the large $N$ limit to the same limit as the distributional limit of a Gaussian tensor model. This generalizes the universality of random matrices to random tensors. We then prove a second, stronger, universality result. Under the weaker assumption that the joint probability distribution of tensor entries is invariant, assuming that the cumulants of this invariant distribution are uniformly bounded, we prove that in the large $N$ limit the tensor again converges in distribution to the distributional limit of a Gaussian tensor model. We emphasize that the covariance of the large $N$ Gaussian isnot universal, but depends strongly on the details of the joint distribution.},
author = {Gurau, Razvan},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {random tensors; large $N$ limit; random tensors; large limit},
language = {eng},
number = {4},
pages = {1474-1525},
publisher = {Gauthier-Villars},
title = {Universality for random tensors},
url = {http://eudml.org/doc/272101},
volume = {50},
year = {2014},
}

TY - JOUR
AU - Gurau, Razvan
TI - Universality for random tensors
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2014
PB - Gauthier-Villars
VL - 50
IS - 4
SP - 1474
EP - 1525
AB - We prove two universality results for random tensors of arbitrary rank $D$. We first prove that a random tensor whose entries are $N^{D}$ independent, identically distributed, complex random variables converges in distribution in the large $N$ limit to the same limit as the distributional limit of a Gaussian tensor model. This generalizes the universality of random matrices to random tensors. We then prove a second, stronger, universality result. Under the weaker assumption that the joint probability distribution of tensor entries is invariant, assuming that the cumulants of this invariant distribution are uniformly bounded, we prove that in the large $N$ limit the tensor again converges in distribution to the distributional limit of a Gaussian tensor model. We emphasize that the covariance of the large $N$ Gaussian isnot universal, but depends strongly on the details of the joint distribution.
LA - eng
KW - random tensors; large $N$ limit; random tensors; large limit
UR - http://eudml.org/doc/272101
ER -

References

top
  1. [1] A. Abdesselam and V. Rivasseau. Trees, forests and jungles: A botanical garden for cluster expansions. In Constructive Physics. V. Rivasseau (Ed). Lecture Notes in Physics 446. Springer, Berlin, 1995. Zbl0822.60095MR1356024
  2. [2] J. Ambjorn, B. Durhuus and T. Jonsson. Three-dimensional simplicial quantum gravity and generalized matrix models. Mod. Phys. Lett. A6 (1991) 1133–1146. Zbl1020.83537MR1115607
  3. [3] G. W. Anderson, A. Guionnet and O. Zeitouni. An Introduction to Random Matrices. Studies in Advanced Mathematics 118. Cambridge Univ. Press, Cambridge, 2009. Zbl1184.15023MR2760897
  4. [4] J. Ben Geloun and V. Rivasseau. A renormalizable 4-dimensional tensor field theory. Comm. Math. Phys. 318 69–109. Available at arXiv:1111.4997 [hep-th]. Zbl1261.83016MR3017064
  5. [5] V. Bonzom, R. Gurau, A. Riello and V. Rivasseau. Critical behavior of colored tensor models in the large limit. Nucl. Phys. B 853 (2011) 174–195. Available at arXiv:1105.3122 [hep-th]. Zbl1229.81222MR2831765
  6. [6] V. Bonzom, R. Gurau and V. Rivasseau. Random tensor models in the large limit: Uncoloring the colored tensor models. Phys. Rev. D 85 (2012) 084037. Available at arXiv:1202.3637 [hep-th]. Zbl1229.81222MR2913765
  7. [7] B. Collins. Moments and cumulants of polynomial random variables on unitary groups, the Itzykson–Zuber integral, and free probability. Int. Math. Res. Not. 17 (2003) 953–982. Available at arXiv:math-ph/0205010. Zbl1049.60091MR1959915
  8. [8] B. Collins and P. Sniady. Integration with respect to the Haar measure on unitary, orthogonal and symplectic group. Comm. Math. Phys. 264 (2006) 773–795. Available at arXiv:math-ph/0402073. Zbl1108.60004MR2217291
  9. [9] A. Connes and D. Kreimer. Hopf algebras, renormalization and noncommutative geometry. Comm. Math. Phys. 199 (1998) 203–242. Available at arXiv:hep-th/9808042. Zbl0932.16038MR1660199
  10. [10] A. Connes and D. Kreimer. Insertion and elimination: The doubly infinite Lie algebra of Feynman graphs. Ann. Henri Poincaré 3 (2002) 411–433. Available at arXiv:hep-th/0201157. Zbl1033.81061MR1915297
  11. [11] F. J. Dyson. Statistical theory of the energy levels of complex systems. I. J. Math. Phys.3 (1962) 140–156. Zbl0105.41604MR143556
  12. [12] F. J. Dyson. Correlations between eigenvalues of a random matrix. Comm. Math. Phys.19 (1970) 235–250. Zbl0221.62019MR278668
  13. [13] L. Erdos. Universality of Wigner random matrices: A survey of recent results. Uspekhi Mat. Nauk 66 (2011) 67–198. Available at arXiv:1004.0861v2 [math-ph]. Zbl1230.82032MR2859190
  14. [14] G. Gallavotti and F. Nicolo. Renormalization theory in four-dimensional scalar fields. I. Comm. Math. Phys.100 (1985) 545–590. MR806252
  15. [15] J. Glimm and A. Jaffe. Quantum Physics. A Functional Integral Point of View, 2nd edition. Springer, New York, 1987. Zbl0461.46051MR887102
  16. [16] M. Gross. Tensor models and simplicial quantum gravity in 2-D. Nucl. Phys. Proc. Suppl. 25A (1992) 144–149. Zbl0957.83511MR1182621
  17. [17] H. Grosse and R. Wulkenhaar. Renormalization of theory on noncommutative in the matrix base. Comm. Math. Phys. 256 (2005) 305–374. Available at arXiv:hep-th/0401128. Zbl1075.82005MR2160797
  18. [18] R. Gurau. Lost in translation: Topological singularities in group field theory. Class. Quant. Grav. 27 (2010) 235023. Available at arXiv:1006.0714 [hep-th]. Zbl1205.83022MR2738259
  19. [19] R. Gurau. The expansion of colored tensor models. Ann. Henri Poincaré 12 (2011) 829–847. Available at arXiv:1011.2726 [gr-qc]. Zbl1218.81088MR2802384
  20. [20] R. Gurau. The complete expansion of colored tensor models in arbitrary dimension. Ann. Henri Poincaré 13 (2011) 399–423. Available at arXiv:1102.5759 [gr-qc]. Zbl1245.81118MR2909101
  21. [21] R. Gurau. Colored group field theory. Comm. Math. Phys. 304 (2011) 69–93. Available at arXiv:0907.2582 [hep-th]. Zbl1214.81170MR2793930
  22. [22] R. Gurau. A generalization of the Virasoro algebra to arbitrary dimensions. Nucl. Phys. B 852 (2011) 592–614. Available at arXiv:1105.6072 [hep-th]. Zbl1229.81129MR2826235
  23. [23] R. Gurau and V. Rivasseau. The expansion of colored tensor models in arbitrary dimension. Europhys. Lett. 95 (2011) 50004. Available at arXiv:1101.4182 [gr-qc]. Zbl1218.81088MR2802384
  24. [24] R. Gurau and J. P. Ryan. Colored tensor models – A review. SIGMA 8 (2012) 020. Available at arXiv:1109.4812 [hep-th]. Zbl1242.05094MR2942819
  25. [25] S. K. Lando, A. K. Zvonkin, R. V. Gamkrelidze and V. A. Vassiliev. Graphs on Surfaces and Their Applications. Encyclopaedia of Mathematical Sciences 141. Springer, Berlin, 2004. Zbl1040.05001MR2036721
  26. [26] J. Martinet and J.-P. Ramis. Elementary acceleration and multisummability I. Ann. Inst. H. Poincaré Phys. Théor.54 (1991) 331–401. Zbl0748.12005MR1128863
  27. [27] M. L. Mehta. Random Matrices. Pure and Applied Mathematics (Amsterdam) 142. Elsevier/Academic Press, Amsterdam, 2004. Zbl1107.15019MR2129906
  28. [28] A. Nica and R. Speicher. Lectures on the Combinatorics of Free Probability. London Mathematical Society Lecture Note Series 335. Cambridge Univ. Press, Cambridge, 2006. Zbl1133.60003MR2266879
  29. [29] L. Pastur and M. Shcherbina. Eigenvalue Distribution of Large Random Matrices. Mathematical Surveys and Monographs 171. Amer. Math. Soc., Providence, RI, 2011. Zbl1244.15002MR2808038
  30. [30] V. Rivasseau. From Perturbative to Constructive Renormalization, 2nd edition. Princeton Univ. Press, Princeton, 1991. MR1174294
  31. [31] V. Rivasseau. Constructive matrix theory. JHEP 0709 (2007) 008. Available at arXiv:0706.1224 [hep-th]. MR2342423
  32. [32] V. Rivasseau and Z. Wang. Loop vertex expansion for theory in zero dimension. J. Math. Phys. 51 (2010) 092304. Available at arXiv:1003.1037 [math-ph]. Zbl1309.81128MR2742808
  33. [33] A. D. Sokal. An improvement of Watson’s theorem on Borel summability. J. Math. Phys.21 (1980) 261–263. Zbl0441.40012MR558468
  34. [34] R. Speicher. Multiplicative functions on the lattice of non-crossing partitions and free convolution. Math. Ann.298 (1994) 611–628. Zbl0791.06010MR1268597
  35. [35] G. ’t Hooft. A planar diagram theory for strong interactions. Nucl. Phys. B 72 (1974) 461–473. 
  36. [36] T. Tao and V. Vu. Random matrices: Universality of local eigenvalue statistics. Acta Math. 206 (1) (2011) 127–204. Available at arXiv:0906.0510v10. Zbl1217.15043MR2784665
  37. [37] D. Voiculescu. Symmetries of some reduced free product -algebras. In Operator Algebras and Their Connections with Topology and Ergodic Theory (Buşteni, 1983) 556–588. Lecture Notes in Mathematics 1132. Springer, New York, 1983. Zbl0618.46048MR799593
  38. [38] D. Voiculescu. Limit laws for random matrices and free products. Invent. Math.104 (1991) 201–220. Zbl0736.60007MR1094052
  39. [39] D. Voiculescu, K. Dykema and A. Nica. Free Random Variables. CRM Monograph Series 1. Amer. Math. Soc., Providence, RI, 1992. Zbl0795.46049MR1217253

NotesEmbed ?

top

You must be logged in to post comments.